Scattering theory for a nonlinear system of wave equations with critical growth

Changxing Miao ¹   ; Youbin Zhu ¹

¹ Institute of Applied Physics and Computational Mathematics P.O. Box 8009 Beijing 100088, China

Colloquium Mathematicum, Tome 106 (2006) no. 1, pp. 69-81

Résumé

We consider scattering properties of the critical nonlinear system of wave equations with Hamilton structure $$ \cases{ u_{tt}-{\mit \Delta } u=-F_1(|u|^2,|v|^2)u,\cr v_{tt}-{\mit \Delta } v=-F_2(|u|^2, |v|^2)v, \cr }$$ for which there exists a function $F(\lambda , \mu )$ such that $$ {\partial F(\lambda ,\mu )\over \partial \lambda }=F_1(\lambda ,\mu ),\hskip 1em {\partial F(\lambda ,\mu )\over \partial \mu }=F_2(\lambda ,\mu ). $$ By using the energy-conservation law over the exterior of a truncated forward light cone and a dilation identity, we get a decay estimate for the potential energy. The resulting global-in-time estimates imply immediately the existence of the wave operators and the scattering operator.

Détail

DOI : 10.4064/cm106-1-6

Keywords: consider scattering properties critical nonlinear system wave equations hamilton structure cases mit delta f mit delta f which there exists function lambda partial lambda partial lambda lambda hskip partial lambda partial lambda using energy conservation law exterior truncated forward light cone dilation identity get decay estimate potential energy resulting global in time estimates imply immediately existence wave operators scattering operator